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Section 8.3: Exploring the Addition of Complex Waves

re1 | im1

re2 | im2

re3 | im3

click to see the real and imaginary parts of the wave functions.

Please wait for the animation to completely load.

In this Exploration we investigate how two and three time-dependent plane waves can be added together to begin to resemble a localized wave packet (in Section 8.4 you can add up to 40 plane waves together).  In the animation, ħ = 2m = 1.  Restart.

  1. With the default settings, explain why the arguments of the cosines and sines are of the form (5*x-25*t) and (-5*x-25*t).  In other words, what does the ±5 signify and what does the 25 signify?  Remember that ħ = 2m = 1 in this animation.
  2. With the default settings, describe the sum of the two plane waves. Look at the real and imaginary parts of the wave functions to verify your conjecture.
  3. Now change wave functions 2 and 3 to re2 = 1*cos(4*x-16*t), im2 = 1*sin(4*x-16*t), re3 = 1*cos(6*x-36*t), and im3 = 1*sin(6*x-36*t). What results?  Now change the number multiplying plane wave 2 and plane wave 3 to 0.5. What wave results now?  How does this superposition accomplish this result?
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